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THE NONLLINEAR MATHEMATICAL 2D MODEL FOR THE ANALYSIS OF TEMPERATURE REGIMES IN THERMOSENSITIVE LAYERED MEDIUM WITH INCLUSIONS
Abstract
The aim of the paper is determination of the temperature field which is caused by a heat flux in a thermosensitive (thermophysical parameters depending on temperature) layered medium which contains a foreign inclusion. The heat flux is concentrated at one of the boundary surfaces of the medium, the other boundary surface of which is thermally insulated. There exists ideal heat contact at the surfaces of the conjugated layers. In order to determine temperature regimes in such a medium, a nonlinear equation of heat conduction with nonlinear boundary conditions is used. In order to solve the nonlinear boundary value problem of heat conduction, we introduce a linearizing function which enables us to obtain a partially linearized differential equation and linear boundary conditions to determine this function. After the piecewise-linear approximation of temperature with respect to spatial coordinates is carry out, a linear differential equation with discontinuous coefficients in the linearizing function is obtained. An analytical-numerical solution of the obtained linear boundary value problem is found with the use of Fourier integral transformation which determines the linearizing function and enables us to obtain calculation formulae for calculation of temperature. Let us consider a linear temperature dependence of the coefficient of heat conductivity of the material for a two-layer medium with an inclusion, and let us make a comparative numerical analysis of the distribution of temperature for a linear (coefficient of heat conductivity of the materials of the layers is a constant quantity) and a nonlinear one (coefficient of heat conductivity of the materials of the layers is a linear variable with respect to temperature) models (materials of the layers are U12 and 08 steels). The temperature field for a layer with a through inclusion (material of the layer is ??94?I ceramics, material of the inclusion is silver) have been calculated and analyzed.
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